on the analytic solution of magnetohydrodynamic flows of non-newtonian fluids over a stretching...

J. Fluid Mech. (2003), vol. 488, pp. 189–212. c© 2003 Cambridge University Press

DOI: 10.1017/S0022112003004865 Printed in the United Kingdom189

On the analytic solution ofmagnetohydrodynamic flows of

non-Newtonian fluids over a stretching sheet

By SHI-JUN LIAOSchool of Naval Architecture and Ocean Engineering, Shanghai Jiao Tong University,

Shanghai 200030, [email protected]

(Received 4 September 2002 and in revised form 25 February 2003)

A powerful, easy-to-use analytic technique for nonlinear problems, the homotopyanalysis method, is employed to give analytic solutions of magnetohydrodynamicviscous flows of non-Newtonian fluids over a stretching sheet. For the so-calledsecond-order and third-order power-law fluids, the explicit analytic solutions are givenby recursive formulas with constant coefficients. Also, for real power-law index andmagnetic field parameter in a quite large range, an analytic approach is proposed. Allof our analytic results agree well with numerical ones. In particular, a simple analyticformula of the dimensionless velocity gradient at the wall is found, which is accuratefor all real power-law indices and magnetic field parameters. This analytic formulacan give sufficiently accurate results for the skin friction on the moving sheet that itwould find wide application in industries. Physically, they indicate that the magneticfield tends to increase the skin friction, and that this effect is more pronounced forshear-thinning than for shear-thickening fluids.

1. IntroductionInvestigations of boundary layer flows of viscous fluid due to a stretching sheet

have important practical applications in chemical and metallurgy industries. Tothe best of the author’s knowledge, it was first studied by Sakiadis (1961) and thenfollowed by many other researchers such as Erickson, Fan & Fox (1966), Crane (1970),McCormack & Crane (1973), McLeod & Rajagopal (1987). The boundary-layer flowsof non-Newtonian fluids due to a stretching plane were investigated by Rajagopal, Na& Gupta (1984, 1987), Troy et al. (1987), Dandapat & Gupta (1989), Andersson &Dandapat (1991). Viscous flows due to a moving sheet in electronic-magnetic fields, i.e.magnetodhydrodynamic (MHD) flows, are relevant to many practical applications inthe metallurgy industry, such as the cooling of continuous strips and filaments drawnthrough a quiescent fluid and the purification of molten metals from non-metallicinclusions. MHD flows of Newtonian fluids were investigated by Pavlov (1974),Chakrabarti & Gupta (1979), Vajravelu (1986), Takhar, Raptis & Perdikis (1987),Andersson (1992, 1995). To the author’s knowledge, MHD flows of non-Newtonianfluids were first studied by Sarpkaya (1961) and then followed by Djukic (1973, 1974),Andersson, Bech & Dandapat (1992), etc.

Consider the flow of an electrically conducting fluid, obeying the power-law modelin the presence of a transverse magnetic field, past a flat sheet lying on the planey = 0, the flow being confined to y > 0. Two equal and opposite forces are applied

190 S.-J. Liao

along the x-axis so that the wall is stretched keeping the origin fixed. The boundarylayer flow is governed by

∂u

∂x+

∂v

∂y= 0, (1)

u∂u

∂x+ v

∂u

∂y=

1

ρ

∂τxy

∂y−

(σB2

0

ρ

)u, (2)

where u and v are velocity components in the x- and y-directions, ρ, σ, B0 and τx,y

are the density, electrical conductivity, magnetic field and shear stress, respectively.The shear stress tensor is defined by the Ostwald–de-Wale model

τij = 2K(2DklDkl)(κ−1)/2Di,j , (3)

where

Dij =1

2

(∂ui

∂xj

+∂uj

∂xi

)denotes the stretching tensor, K is the consistency coefficient and κ is the power-lawindex. The boundary conditions are

u = Cx, v = 0 at y = 0 (4)

and

u → 0 as y → +∞, (5)

where C is a positive constant. Under the transformation

ψ =

(K/ρ

C1−2κ

)1/(κ+1)

x2κ/(κ+1)F (ξ ) (6)

and

ξ = y

(C2−κ

K/ρ

)1/(κ+1)

x(1−κ)/(1+κ), (7)

where

u =∂ψ

∂y, v = −∂ψ

∂x

define the stream function ψ , the governing equation becomes

κ [−F ′′(ξ )]κ−1 F ′′′(ξ ) +

(2κ

κ + 1

)F (ξ ) F ′′(ξ ) − F ′2(ξ ) − M F ′(ξ ) = 0, (8)

subject to the boundary conditions

F (0) = 0, F ′(0) = 1, F ′(+∞) = 0, (9)

where the prime denotes differentiation with respect to ξ and M = σB20/(ρC) is the

magnetic parameter. The skin friction coefficient Cf at the wall is given by

Cf =τw

12ρ(Cx)2

= 2[−F ′′(0)]κ[(Cx)2−κxκ

K/ρ

]−1/(1+κ)

, (10)

where (Cx)2−κxκ/(K/ρ) is the local Reynolds number based on the sheet velocity Cx.For details refer to Andersson et al. (1992).

Andersson et al. (1992) numerically solved the nonlinear differential equations (8)and (9) for seven values of the power-law index in the range 0.4 � κ � 2 and

Analytic solution of MHD flows of non-Newtonian fluids 191

for five different values of the magnetic field 0 � M � 2. However, to the bestof the author’s knowledge, no analytic results have been reported for κ and M ingeneral cases. In this paper, an analytic tool for nonlinear problems, namely thehomotopy analysis method (HAM) (Liao 1992), is employed to solve the nonlineardifferential equations (8) and (9). The homotopy analysis method is based on a basicconcept in topology, i.e. homotopy (Hilton 1953) that is widely applied in numericaltechniques (Alexander & Yorke 1978; Chan & Keller 1982; Dinar & Keller 1985;Grigolyuk & Shalashilin 1991). Unlike perturbation techniques (Cole 1958; Van Dyke1975; Hinch 1991; Murdock 1991; Nayfeh 2000), the homotopy analysis method isindependent of small/large parameters. Unlike all other reported perturbation andnon-perturbation techniques such as the artificial small parameter method (Lyapunov1892), the δ-expansion method (Karmishin, Zhukov & Kolosov 1990; Awrejcewicz,Andrianov & Manevitch 1998) and Adomian’s (1976, 1994) decomposition method,the homotopy analysis method provides us with a simple way to adjust and control theconvergence region and rate of approximation series. The homotopy analysis methodhas been successfully applied to many nonlinear problems such as viscous flows(Liao 1999a, b), heat transfer (Liao & Campo 2002; Wang et al. 2003), nonlinearoscillations (Liao & Chwang 1998), nonlinear water waves (Liao & Cheung 2003),Thomas-Fermi’s atom model (Liao 2003), etc. In particular, by means of the homotopyanalysis method, the author (Liao 2002) gave a drag formula for a sphere in auniform stream, which agrees well with experimental results in a considerably largerregion of Reynolds number than those of all reported analytic drag formulas. All ofthese successful applications of the homotopy analysis method verify its validity fornonlinear problems in science and engineering.

In this paper we now employ the homotopy analysis method to solve the MHDviscous flows of non-Newtonian fluids and propose analytic solutions of (8) and (9)for κ > 0 and 0 � M � 1000.

2. Analytic solutions for integer power-law indexPhysically, the power-law index κ is a positive real number. First, let us consider

the case that κ is a positive integer. When κ = 1, equation (8) becomes

F ′′′(ξ ) + F (ξ ) F ′′(ξ ) − F ′2(ξ ) − M F ′(ξ ) = 0. (11)

This equation has an exact analytic solution (see Pavlov 1974)

F (ξ ) =1 − exp(−

√1 + M ξ )√

1 + M, (12)

which gives

F ′(ξ ) = exp(−√

M + 1 ξ ) (13)

and

F ′′(0) = −√

1 + M. (14)

When κ = 2, equation (8) is

−2 F ′′(ξ ) F ′′′(ξ ) +(

43

)F (ξ ) F ′′(ξ ) − F ′2(ξ ) − M F ′(ξ ) = 0, (15)

and when κ = 3,

3[F ′′(ξ )]2 F ′′′(ξ ) +(

32

)F (ξ ) F ′′(ξ ) − F ′2(ξ ) − M F ′(ξ ) = 0. (16)

192 S.-J. Liao

The zero-order deformation equation

Let λ denote a positive constant. Due to the boundary conditions (9) and the knownexact solution (12) for κ = 1, F (ξ ) for any a positive integer κ can be expressed by aset of base functions

{exp(−n λ ξ ) | n � 0} (17)

in the following form:

F (ξ ) = a0 +

+∞∑n=1

an exp(−n λ ξ ), (18)

where an is a coefficient. Then, from the above expression and the boundary conditions(9), it is straightforward to choose

F0(ξ ) =[1 − exp(−λ ξ )]

λ(19)

as the initial guess of F (ξ ). Also, due to (18) and the governing equation (8), wechoose

L[Φ(ξ ; q)] =∂3Φ(ξ ; q)

∂ξ 3− λ2 ∂Φ(ξ ; q)

∂ξ, (20)

as our auxiliary linear operator, where q is an embedding parameter. Note that theauxiliary linear operator L has the property

L [C1 + C2 exp(−λ ξ ) + C3 exp(λ ξ )] = 0. (21)

Furthermore, we define using (8) the nonlinear operator

N [Φ(ξ ; q)] = κ

[−∂2Φ(ξ ; q)

∂ξ 2

]κ−1∂3Φ(ξ ; q)

∂ξ 3+

(2κ

κ + 1

)Φ(ξ ; q)

∂2Φ(ξ ; q)

∂ξ 2

−[∂Φ(ξ ; q)

∂ξ

]2

− M∂Φ(ξ ; q)

∂ξ, (22)

where κ is a positive integer. Then, we construct the zero-order deformation equation

(1 − q) L [Φ(ξ ; q) − F0(ξ )] = � q exp(−lλ ξ ) N [Φ(ξ ; q)] , q ∈ [0, 1], (23)


Φ(0; q) = 0,∂Φ(ξ ; q)

∂ξ

∣∣∣∣ξ=0

= 1,∂Φ(ξ ; q)

∂ξ

∣∣∣∣ξ=+∞

= 0. (24)

When q = 0, it is straightforward to show that the solution of (23) and (24) is

Φ(ξ ; 0) = F0(ξ ). (25)

When q = 1, equations (23) and (24) are respectively the same as (8) and (9), provided

Φ(ξ ; 1) = F (ξ ). (26)

Thus, as q increases from 0 to 1, Φ(ξ ; q) varies from the initial guess F0(ξ ), definedby (19), to the exact solution F (ξ ) governed by (8) and (9). This kind of continuousvariation is called deformation in topology.


Using (25), it is straightforward to expand Φ(ξ ; q) in a power series of the embeddingparameter q as follows:

Φ(ξ ; q) = F0(ξ ) +

+∞∑m=1

Fm(ξ ) qm, (27)

where

Fm(ξ ) =1

m!

∂mΦ(ξ ; q)

∂qm

∣∣∣∣q=0

. (28)

Note that the zero-order deformation equation (23) contains a non-zero auxiliaryparameter �. Thus, Φ(ξ ; q) and Fm(ξ ) are dependent upon this parameter. Obviously,� also affects the convergence rate and region of the series (27). Assume that � ischosen so that the series (27) is convergent at q = 1. Then, due to (26) and (27), wehave the relationship

F (ξ ) = F0(ξ ) +

+∞∑m=1

Fm(ξ ). (29)

The high-order deformation equation

Differentiating the zero-order deformation equations (23) and (24) m times withrespect to q and then dividing them by m! and finally setting q = 0, we have, due tothe definition (28), the mth-order deformation equation

L [Fm(ξ ) − χm Fm−1(ξ )] = � exp(−lλ ξ ) Rm(ξ ), (30)


Fm(0) = 0, F ′m(0) = 0, F ′

m(+∞) = 0, (31)

where

Rm(ξ ) =1

(m − 1)!

∂m−1N[Φ(ξ ; q)]

∂qm−1

∣∣∣∣q=0

(32)

and

χm =

{0 when m � 1,

1 when m � 2.(33)

Note that Rm(ξ ) is dependent upon the integer power-law index κ . When κ = 1, wehave

Rm(ξ ) = F ′′′m−1(ξ ) +

m−1∑n=0

[Fn(ξ ) F ′′

m−1−n(ξ ) − F ′n(ξ ) F ′

m−1−n(ξ )]

− M F ′m−1(ξ ). (34)

When κ = 2,

Rm(ξ ) =

m−1∑n=0

[4Fn(ξ ) F ′′

m−1−n(ξ )/3 − F ′n(ξ ) F ′

m−1−n(ξ )]

− MF ′m−1(ξ ) − 2

m−1∑n=0

F ′′n (ξ ) F ′′′

m−1−n(ξ ). (35)

194 S.-J. Liao

When κ = 3,

Rm(ξ ) =

m−1∑n=0

[3Fn(ξ ) F ′′

m−1−n(ξ )/2 − F ′n(ξ ) F ′

m−1−n(ξ )]

− M F ′m−1(ξ )

+ 3

m−1∑n=0

n∑j=0

F ′′j (ξ ) F ′′

n−j (ξ ) F ′′′m−1−n(ξ ). (36)

Note also that Rm(ξ ) contains the term exp(−λ ξ ). It is found that, when l = 0, theright-hand side of equation (30) contains the term exp(−λ ξ ). Then, due to (21), thesolution of equation (30) has the term

ξ exp(−λ ξ ),

which disobeys expression (18) however. So, in order to obey (18), we have to choosel � 1. On the other hand, it is found that, when l � 2, the solution of (30) does notcontain the term

exp(−2λ ξ ),

so that the coefficient of the term exp(−2λ ξ ) cannot be further improved even ifm → +∞. To avoid both these difficulties, we must have

l = 1. (37)

It should be emphasized that we still have freedom to choose a positive value forλ and a negative value for the auxiliary parameter �. For simplicity we choose λ = 1in this section for the integer power-law index κ . We will show that the auxiliaryparameter � plays an important role in the homotopy analysis method.

2.1. Solutions when κ = 1

The exact solution (12) can be employed to verify the validity of the foregoing analyticapproach. It is straightforward to solve the linear equations (30) and (31). It is foundthat, when κ = 1, Fm(ξ ) governed by (30) and (31) can be expressed by

Fm(ξ ) =

2m+1∑n=0

am,n exp(−n λ ξ ), (38)

where am,n is a coefficient. Substituting (38) into the mth-order deformationequations (30) and (31), we have the recursive formulas

am,n = χm χ2m+1−n am−1,n +� χ2m+2−n (n − 1) [λ2 (n − 1)2 − M] am−1,n−1

λ2n (n2 − 1)

− � (Am,n−1 − χn−1 Bm,n−1)

λ n ( n2 − 1)(39)

for 2 � n � 2m + 1 and

am,1 = −2m+1∑n=2

n am,n, (40)

am,0 = −2m+1∑n=1

am,n (41)


under the definitions

Am,j =

m−1∑n=0

min{2n+1,j−1}∑i=max{0,j−2m+2n+1}

(j − i)2 an,i am−n−1,j−i , 1 � j � 2m (42)

and

Bm,j =

m−1∑n=0

min{2n+1,j−1}∑i=max{1,j−2m+2n+1}

i(j − i) an,i am−n−1,j−i , 2 � j � 2m. (43)

Due to the initial guess (19), we have the first two coefficients:

a0,0 = 1/λ, a0,1 = −1/λ. (44)

Thus, starting from these two coefficients, we can calculate the coefficients am,n form = 1, 2, 3, . . . , 0 � n � 2m + 1 by means of the above recursive formulas. So, wehave the explicit analytic solution

F (ξ ) =

+∞∑k=0

2k+1∑n=0

ak,n exp(−n λ ξ ) (45)

for the Newtonian fluid. At the mth-order of approximation we have

F (ξ ) ≈m∑

k=0

2k+1∑n=0

ak,n exp(−n λ ξ ), (46)

which gives

F ′(ξ ) ≈ −λ

m∑k=0

2k+1∑n=1

n ak,n exp(−n λ ξ ), (47)

F ′′(0) ≈ λ2

m∑k=0

2k+1∑n=1

n2 ak,n. (48)

It is found that when κ = 1 the mth-order approximation of F ′′(0) can be expressedby

F ′′(0) ≈m∑

n=0

αm,n1 Mn, (49)

where αm,n1 is coefficient dependent upon �. Thus, our results contain an auxiliary

parameter �, which plays an important role in the homotopy analysis method. Asverified in the author’s previous publications (Liao & Chwang 1998; Liao 1999 a, b,2003; Liao & Campo 2002; Liao & Cheung 2003), it is the value of the auxiliaryparameter � which can adjust and control the convergence region and rate of theseries given by the homotopy analysis method. It is found that the convergence regionof above series increases as � tends to zero from below, as shown in figure 1.

A power series of F ′′(0) can be obtained by expanding the exact solution (14) asfollows:

F ′′(0) ≈ 1 +M

2− M2

8+

M3

16− 5M4

128+ · · · , (50)

which is convergent in a rather restricted region 0 � M < 2 however, as shown infigure 1. Employing the traditional Pade technique to the above power series of the

196 S.-J. Liao

0 10 20 30 40 50

M

1

2

3

4

5

6

7

8

–F ��(0)

Figure 1. Comparison of F ′′(0) at the 20th-order of approximation given by the homotopyanalysis method with the exact solution (14) and perturbation result (50) when κ = 1.Symbols: exact solution (14); solid line: perturbation result (50); dashed line: homotopyanalysis approximation when � = −1; dash-dotted line: homotopy analysis approximationwhen � = −1/2; dash-dot-dotted line: homotopy analysis approximation when � = −1/4.

physical parameter M , one has the [m, m] Pade approximant

F ′′(0) ≈ −1 +

m∑n=1

δm,n Mn

1 +m∑

n=1

δm,m+n Mn

, (51)

where δm,n is constant. Recently, Liao & Cheung (2003) proposed a so-called homotopy-Pade method, which is more efficient than the traditional Padeapproximant. From (27), we have the (2m)th-order approximation

∂2Φ(ξ ; q)

∂ξ 2

∣∣∣∣ξ=0

≈2m∑n=0

F ′′n (0) qn. (52)

Employing the traditional Pade technique to the power series of the embeddingparameter q , we have the [m, m] Pade approximant

∂2Φ(ξ ; q)

∂ξ 2

∣∣∣∣ξ=0

≈ −1 +

m∑n=1

γm,n qn

1 +m∑

n=1

γm,m+n qn

, (53)


0 25

M

1

–F ��(0)

50 75 100

2

3

4

5

6

7

8

9

10

11

Figure 2. Comparison of F ′′(0) given by the homotopy-Pade method with the exact solution(14) and the traditional Pade technique when κ = 1. Filled circles: exact solution (14);open circles: traditional [10,10] Pade approximation; dashed line: [4,4] homotopy-Padeapproximation; solid line: [6,6] homotopy-Pade approximation.

where γm,n is a coefficient. Setting q = 1 in above series, we obtain using (29) that

F ′′(0) ≈ −1 +

m∑n=1

γm,n

1 +m∑

n=1

γm,m+n

. (54)

It is found that the above [m, m] homotopy-Pade expression can be rewritten as

F ′′(0) ≈ −1 +

m2∑n=1

m,n1 Mn

1 +m2∑n=1

m,m2+n1 Mn

, (55)

where m,n1 (1 � n � 2m2) is constant, which is found to be independent of the

auxiliary parameter �. Note that the traditional Pade approximant analyses resultsexpressed by physical parameters, which have been obtained by setting q = 1.However, in the homotopy-Pade method, one first analyses the power series in the em-bedding parameter q and then sets q = 1. This is the distinction between the tra-ditional Pade approximant and the homotopy-Pade ones. It is interesting that theabove [m, m] homotopy-Pade approximant contains the term Mm2

while the traditional[m, m] Pade approximant (51) has only the term Mm. As shown in figure 2, even

198 S.-J. Liao

n Pade approximant Homotopy-Pade approximant

1 2.42857 3.083332 3.03390 3.301343 3.23265 3.316054 3.92217 3.316615 3.30958 3.316626 3.31459 3.316628 3.31646 3.31662

10 3.31661 3.31662

Table 1. Comparison of the [n, n] homotopy-Pade approximant of −F ′′(0) with thetraditional [n, n] Pade approximant when κ = 1 and M = 10.

0 0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

�

F (�)

F �(�)

Figure 3. Comparison of F (ξ ) and F ′(ξ ) given by the homotopy-Pade technique with theexact solution (12) and (13) when κ = 1 and M = 10. Filled circles: exact solution (12); opencircles: exact solution (13); solid line: [5,5] homotopy-Pade approximation of F (ξ ); dashedline: [5,5] homotopy-Pade approximation of F ′(ξ ).

the [4,4] homotopy-Pade approximant gives as accurate an approximation as thetraditional [10, 10] Pade approximant and the [6,6] homotopy-Pade approximantagrees well with the exact result. Clearly, the homotopy-Pade approximant of −F ′′(0)converges much faster to the exact result

√1 + M than the traditional Pade technique,

as shown in table 1 and figures 1 and 2. Similarly, one can employ the homotopy-Pade technique to obtain more accurate results of F (ξ ) and F ′(ξ ). For example, whenM = 10, the [5,5] homotopy-Pade approximants of F (ξ ) and F ′(ξ ) agree well with theexact solutions (12) and (13), respectively, as shown in figure 3. It is interesting that


the numerator and denominator of the homotopy-Pade approximants of F (ξ ) andF ′(ξ ) are sums of exponential functions of ξ . All of these clearly verify the validity ofthe proposed analytic approach and the homotopy-Pade method, although the exactsolution is known when κ = 1.


For the second-order power-law fluid (κ = 2), it is found that Fm(ξ ) governed by (30)and (31) can be expressed by

Fm(ξ ) =

2m+1∑n=0

am,n exp(−n λ ξ ). (56)

Substituting this into the mth-order deformation equations (30) and (31), we have therecursive formula

am,n = χm χ2m+1−n am−1,n − � χ2m+2−n M am−1,n−1

λ2n (n + 1)

− � (4Am,n−1/3 − χn−1 Bm,n−1)

λ n (n2 − 1)− 2 � λ2 χn−1 Cm,n−1

n (n2 − 1)(57)

for 2 � n � 2m + 1, and

am,1 = −2m+1∑n=2

n am,n, (58)

am,0 = −2m+1∑n=1

am,n, (59)

where

Cm,j =

m−1∑n=0

min{2n+1,j−1}∑i=max{1,j−2m+2n+1}

i2(j − i)3 an,i am−n−1,j−i , 2 � j � 2m (60)

and Am,n, Bm,n are defined by (42) and (43), respectively. So, we have the explicitanalytic solution

F (ξ ) =

+∞∑k=0

2k+1∑n=0

ak,n exp(−n λ ξ ) (61)

for the second-order power-law fluid. The mth-order approximations of F (ξ ) andF ′(ξ ) are the same as (46) and (47), respectively. The first two coefficients are givenby (44).


F ′′(0) ≈m∑

n=0

αm,n2 Mn, (62)

where αm,n2 is a coefficient dependent upon �. Similarly, as � tends to zero from below,

the convergence region of F ′′(0) enlarges. The analytic approximations of F ′′(0) forM = 1/2, 1, 3/2 and 2 when � = −1/2 are given in table 2, and they agree well withthe numerical results 1.093, 1.187, 1.269 and 1.342 given by Andersson et al. (1992),respectively. It is found that when � = −1/(2 + M/10) the series of our analyticsolutions converge for any a magnetic field parameter M � 0.

200 S.-J. Liao

m M = 1/2 M = 1 M = 3/2 M = 2

5 1.054 1.157 1.247 1.32510 1.070 1.173 1.260 1.33620 1.082 1.182 1.266 1.34030 1.087 1.185 1.268 1.34140 1.090 1.186 1.269 1.34150 1.091 1.187 1.269 1.34260 1.092 1.187 1.269 1.34270 1.093 1.187 1.269 1.34280 1.093 1.187 1.269 1.342

Table 2. The mth-order homotopy analysis solution for −F ′′(0) when κ = 2 and � = −1/2.

n M = 1/2 M = 1 M = 3/2 M = 2

5 1.090 1.185 1.266 1.33910 1.094 1.187 1.269 1.34220 1.092 1.187 1.269 1.34230 1.093 1.187 1.269 1.34240 1.093 1.187 1.269 1.342

Table 3. The [n, n] homotopy-Pade approximant of −F ′′(0) when κ = 2.

It is straightforward to employ the traditional Pade technique to the series of F ′′(0).It is found that this cannot enlarge convergence regions of the series (62), which aremainly determined by �. Thus, the traditional Pade technique seems of no use whenκ = 2.

Unlike the traditional Pade technique, the homotopy-Pade method can greatlyincrease the convergence rate and region of the series of F ′′(0). It is found thatwhen κ = 2 the corresponding [m, m] homotopy-Pade approximant of F ′′(0) can beexpressed by

F ′′(0) ≈ m,0

2 +m2+m∑n=1

m,n2 Mn

1 +m2+m∑n=1

m,m2+m+n2 Mn

, (63)

where the constant m,n2 (0 � n � 2m2 + 2m) is also independent of �. Note that,

unlike (55), the above expression contains the term Mm2+m. By comparing table 2with table 3, it is clear that the convergence rate can be considerably increased by thehomotopy-Pade method. Also, the convergence region can be greatly enlarged. The[5,5] and [6,6] homotopy-Pade approximations of F ′′(0) agree well with numericalresults in the region 0 � M � 1000, respectively, as shown in figure 4. The convergentanalytic results of F ′′(0) agree well with numerical results, as shown in table 4. Thisfurther verifies the validity of our analytic approach.


For the third-order power-law fluid (κ = 3), it is found that Fm(ξ ) governed by (30)and (31) can be expressed by

Fm(ξ ) =

3m+1∑n=0

bm,n exp(−n λ ξ ). (64)


M Analytic result Numerical result Results given by (81)

0 0.980 0.980 0.9690.5 1.093 1.093 1.0871 1.187 1.187 1.184

1.5 1.269 1.269 1.2672 1.342 1.342 1.3403 1.467 1.468 1.4674 1.575 1.575 1.5755 1.670 1.670 1.6706 1.755 1.755 1.7558 1.904 1.904 1.90510 2.033 2.033 2.03320 2.514 2.515 2.51540 3.138 3.138 3.13860 3.580 3.581 3.58080 3.934 3.935 3.934100 4.234 4.234 4.234200 5.324 5.325 5.324400 6.701 6.702 6.701600 7.668 7.670 7.668800 8.439 8.441 8.4391000 9.089 9.091 9.089

Table 4. Comparison of the numerical results with the analytic results of −F ′′(0) given bythe homotopy-Pade technique when κ = 2.

100

M

1

2

3

4

5

6

7

8

–F ��(0)

101 102 103

9

10

Figure 4. Comparison of F ′′(0) given by the homotopy-Pade method with the numericalresults when κ = 2. Solid line: numerical solution; open circle: [5,5] homotopy-Padeapproximation; filled circle: [6,6] homotopy-Pade approximation.

202 S.-J. Liao

Substituting this into the mth-order deformation equations (30) and (31), we have therecursive formula

bm,n = χm χ3m−n bm−1,n − � χ3m+1−n M bm−1,n−1

λ2n ( n + 1)

− � χ3m+2−n (3Dm,n−1/2 − χn−1 Sm,n−1)

λ n (n2 − 1)+

3 � λ4 χn−2 Qm,n−1

n (n2 − 1)(65)

for 2 � n � 3m + 1, where

Dm,j =

m−1∑n=0

min{3n+1,j−1}∑i=max{0,j−3m+3n+2}

(j − i)2 bn,i bm−n−1,j−i , 1 � j � 3m − 1, (66)

Sm,j =

m−1∑n=0

min{3n+1,j−1}∑i=max{1,j−3m+3n+2}

i (j − i) bn,i bm−n−1,j−i , 2 � j � 3m − 1, (67)

Qm,j =

m−1∑n=0

min{3n+2,j−1}∑i=max{2,j−3m+3n+2}

(j − i)3 bm−n−1,j−i Tn,i , 3 � j � 3m (68)

and

Tk,j =

k∑n=0

min{3n+1,j−1}∑i=max{1,j−3k+3n−1}

i2 (j − i)2 bn,i bk−n,j−i , 2 � j � 3k + 2. (69)

Also, from the boundary conditions (31), we have

bm,1 = −3m+1∑n=2

n bm,n, (70)

bm,0 = −3m+1∑n=1

bm,n. (71)

The initial guess (19) gives the first two coefficients as

b0,0 = 1/λ, b0,1 = −1/λ. (72)

Starting from these two coefficients, we can calculate the coefficients bm,n for m =1, 2, 3, · · · , 0 � n � 3m + 1 by means of the above recursive formulas. So, we havethe explicit analytic solution

F (ξ ) =

+∞∑k=0

3k+1∑n=0

bk,n exp(−n λ ξ ) (73)

for the third-order power-law fluid. At the mth-order of approximation we have

F (ξ ) ≈m∑

k=0

3k+1∑n=0

bk,n exp(−n λ ξ ), (74)

which gives

F ′(ξ ) ≈ −λ

m∑k=0

3k+1∑n=1

n bk,n exp(−n λ ξ ) (75)


and

F ′′(0) ≈ λ2

m∑k=0

3k+1∑n=1

n2 bk,n. (76)


F ′′(0) ≈m∑

n=0

αm,n3 Mn, (77)

where αm,n3 is a coefficient dependent upon �. Similarly, as � tends to zero from below,

the convergence region of F ′′(0) enlarges. It is found that the series converges forany M � 0 when � = −1/(3 + M) and that the convergence region of (77) cannotbe enlarged by the traditional Pade technique. When κ = 3, the corresponding [m, m]homotopy-Pade approximant of F ′′(0) can be expressed by

F ′′(0) ≈ m,0

3 +m2+m∑n=1

m,n3 Mn

1 +m2+m∑n=1

m,m2+m+n3 Mn

, (78)

where the constant m,n3 (0 � n � 2m2 + 2m) is independent of �. The convergence

region of F ′′(0) can be greatly enlarged by the homotopy-Pade method, as shown infigure 5. Note that the [10,10] homotopy-Pade approximant of F ′′(0) agrees well withthe numerical results in a larger region than the [5,5] homotopy-Pade approximant. Itis found that the [25,25] homotopy-Pade approximants of F ′′(0) converge in the region0 � M � 1000. The convergent analytic results of F ′′(0) in the range 0 � M � 1000agree well with numerical results, as shown in table 5.

2.4. A simple approximate formula for skin friction

From the definition (10), the skin friction coefficient Cf is determined by thedimensionless velocity gradient F ′′(0) at the wall. When κ = 1, we have the exactformula (14) for F ′′(0), which indicates that, for large M , ln[−F ′′(0)] is positivelyproportional to lnM . Our analytic results indicate that, when κ = 2 and κ = 3,ln[−F ′′(0)] is also positively proportional to lnM , if M is large enough. This impliesthat

F ′′(0) ≈ − (α + β M)γ (79)

might be a good approximation of F ′′(0) for any a positive integer κ � 1. Using theexact solution (14), we have

α = 1, β = 1, γ = 1/2, when κ = 1.

To ensure that (79) is consistent with the exact formula (14) for κ = 1, we first simplyset α = 1 for all κ . Keeping this in mind and plotting curves defined by (79), andcomparing them with analytic results listed in tables 4 and 5, we find, some whatsurprisingly, that, if

α = 1, β =κ + 1

2κ, γ =

1

κ + 1, (80)

the formula (79) is a very good approximation for F ′′(0) when κ = 2 and κ = 3,especially for large M . Furthermore, it is found that even better results for F ′′(0) are

204 S.-J. Liao

M Analytic result Numerical result Result given by (81)

0 0.985 0.985 0.9750.5 1.060 1.060 1.0551 1.123 1.123 1.119

1.5 1.177 1.177 1.1752 1.224 1.224 1.2233 1.306 1.306 1.3054 1.374 1.375 1.3755 1.434 1.435 1.4356 1.488 1.488 1.4888 1.579 1.580 1.58010 1.658 1.658 1.65920 1.942 1.942 1.94240 2.291 2.291 2.29160 2.529 2.529 2.52980 2.713 2.714 2.714100 2.867 2.867 2.867200 3.404 3.404 3.404400 4.044 4.045 4.044600 4.475 4.475 4.475800 4.807 4.808 4.8081000 5.083 5.084 5.083

Table 5. Comparison of the numerical results with the analytic results for −F ′′(0) given bythe homotopy-Pade technique when κ = 3.

100

M

1.0

–F ��(0)

101 102

1.5

2.0

2.5

3.0

Figure 5. Comparison of F ′′(0) given by the homotopy-Pade method with the numericalresults when κ = 3. Solid line: numerical solution; open circle: [5,5] homotopy-Padeapproximation; filled circle: [10,10] homotopy-Pade approximation.


100

M

1

–F ��(0)

101 102 103

2

3

4

5

6

7

8

9

10

� = 2

� = 3

Figure 6. Comparison of F ′′(0) from the approximate expression (81) with the numericalresults. Dashed line: result given by (81) when κ = 2; solid line: result given by (81) when κ = 3;open circle: the numerical results when κ = 2; square: the numerical results when κ =3.

given by choosing

α =

(κ + 1

2κ

)1/(1+κ)

,

i.e.

F ′′(0) ≈ −[(

κ + 1

2κ

)1/(1+κ)

+

(κ + 1

2κ

)M

]1/(κ+1)

, (81)

as shown in table 4, table 5 and figure 6. When κ = 2 and κ = 3, the relative error ofthe above simple expression is less than 0.2% for M � 2. When M � 2, the series ofapproximations converge in most case, provided

λ =

[(κ + 1

2κ

)1/(1+κ)

+

(κ + 1

2κ

)M

]1/(1+κ)

(82)

and

� = −[(

κ + 1

2κ

)1/(1+κ)

+

(κ + 1

2κ

)M

]−1

. (83)

In the next section we will show that the simple analytic expression (81) is valid evenfor real power-law index κ > 0.

206 S.-J. Liao

3. Analytic solutions for real power-law indexThe zero-order deformation equation

For real power-law index κ > 0, F (ξ ) can be expressed by a set of base functions

{ξm exp(−n λ ξ ) | m � 0, n � 0} (84)

in the following form:

F (ξ ) = a0 +

+∞∑n=1

+∞∑m=0

cn,m ξm exp(−n λ ξ ), (85)

where cn is a coefficient. From this expression and the boundary conditions (9), it isstraightforward to choose the same initial guess F0(ξ ) as (19) and the same auxiliarylinear operator as (20) with the same property (21).

Define

κ − 1 = µ + ε, (86)

where µ � 0 is an integer and |ε| < 1 is a real number. Unlike in § 2, the nonlinearoperator N is now defined by

N [Φ(ξ ; q)] = κ

[−∂2Φ(ξ ; q)

∂ξ 2

]µ+ε q∂3Φ(ξ ; q)

∂ξ 3+

(2κ

κ + 1

)Φ(ξ ; q)

∂2Φ(ξ ; q)

∂ξ 2

−[∂Φ(ξ ; q)

∂ξ

]2

− M∂Φ(ξ ; q)

∂ξ, (87)

where κ is a positive real number. Then, we construct the zero-order deformationequations

(1 − q) L [Φ(ξ ; q) − F0(ξ )] = � q exp(−lλ ξ ) N [Φ(ξ ; q)] , q ∈ [0, 1], (88)

subject to the boundary conditions (24).

The high-order deformation equation

Similarly, the solution F (ξ ) is given by

F (ξ ) = F0(ξ ) +

+∞∑m=1

Fm(ξ ), (89)

where Fm(ξ ) is governed by the mth-order deformation equation

L [Fm(ξ ) − χm Fm−1(ξ )] = � exp(−lλ ξ ) Rm(ξ ), (90)

subject to the boundary conditions (31), where

Rm(ξ ) =1

(m − 1)!

∂m−1N[Φ(ξ ; q)]

∂qm−1

∣∣∣∣q=0

= (−1)µ κ

m−1∑n=0

Hµ,m−1−n(ξ ) Wn(ξ ) − M F ′m−1(ξ )

+

m−1∑n=0

[(2κ

κ + 1

)Fn(ξ ) F ′′

m−1−n(ξ ) − F ′n(ξ ) F ′

m−1−n(ξ )

](91)


using the definitions

Hµ,n(ξ ) =1

n!

{∂n

∂qn

(∂3Φ(ξ ; q)

∂ξ 3

[∂2Φ(ξ ; q)

∂ξ 2

]µ)}∣∣∣∣q=0

(92)

and

Wn(ξ ) =1

n!

{∂n

∂qn

[−∂2Φ(ξ ; q)

∂ξ 2

]ε q}∣∣∣∣q=0

. (93)

From the definition (92), we have

H0,n(ξ ) = F ′′′n (ξ ), (94)

H1,n(ξ ) =

n∑i=0

F ′′′i (ξ ) F ′′

n−i(ξ ) =

n∑i=0

H0,i(ξ ) F ′′n−i(ξ ), (95)

H2,n(ξ ) =

n∑i=0

H1,i(ξ ) F ′′n−i(ξ ), (96)

...

which gives the recursive formula

Hµ,n(ξ ) =

n∑i=0

Hµ−1,i(ξ ) F ′′n−i(ξ ), µ � 1. (97)

From the definition (93), we have

W0(ξ ) = 1,

W1(ξ ) = ε ln[−F ′′0 (ξ )],

W2(ξ ) =ε F ′′

1 (ξ )

F ′′0 (ξ )

+

(ε2

2

)ln2[−F ′′

0 (ξ )],

...

Note that Wn(ξ ) for n > 0 can be given by

Wn(ξ ) =

n−1∑i=0

(1 − i

n

)Wi(ξ ) Zn−i(ξ ), (98)

where

Zn(ξ ) =1

n!

{∂n

∂qn

[ε q ln

(−∂2Φ(ξ ; q)

∂ξ 2

)]}∣∣∣∣q=0

(99)

is easier to calculate than Wn(ξ ) defined by (93). Note that, from the definition (19)of F0(ξ ), we have

ln[−F ′′0 (ξ )] = ln λ − λ ξ.

This is the reason why we choose (84) as the base functions of F (ξ ) for real power-lawindex κ . Note that when κ � 1 we can always find integer µ � 0 and a real number|ε| � 1/2 such that the expression (86) holds.

208 S.-J. Liao

n Homotopy-Pade approximant of −F ′′(0)

1 1.54282 1.54543 1.54474 1.54426 1.54418 1.5441

10 1.5441

Table 6. The [n, n] homotopy-Pade approximant of −F ′′(0) when κ = 4/5 and M = 1 withλ = 3/2.

0

0.25

�

F �(�)

1 2 3 4

0.50

0.75

1.00

Figure 7. Comparison of the numerical solution with the analytic approximation of F ′(ξ )given by the homotopy analysis method for real κ when κ = 4/5,M = 1 with λ = 3/2 and� = −8/17. Solid line: numerical solution; open circle: 10th-order analytic approximations;filled circle: 20th-order analytic approximations.

Similarly, to ensure that F (ξ ) can be expressed by (85) and that each coefficientcm,n can be modified as the order of approximation tends to infinity, we set

l = 0

in (88) and (90).To verify the validity of the above approach for any real power-law index κ , let

us first consider the case κ = 1 and M = 1, corresponding to µ = ε = 0. When� = −1/2 and λ = 2, the corresponding homotopy-Pade approximant of F ′′(0)


κ = 4/5 κ = 3/2 κ = 4

M HAM formula (81) HAM formula (81) HAM formula (81)

0 1.028 1.037 0.982 0.971 0.989 0.9811/2 1.308 1.312 1.131 1.126 1.046 1.0411 1.544 1.547 1.257 1.255 1.093 1.090

3/2 1.754 1.756 1.367 1.366 1.133 1.1312 1.945 1.947 1.466 1.465 1.167 1.1673 2.289 2.290 1.638 1.637 1.228 1.2275 2.874 2.875 1.918 1.918 1.322 1.32210 4.035 4.035 2.436 2.436 1.482 1.48225 6.517 6.517 3.428 3.428 1.752 1.75250 9.480 9.480 4.485 4.485 2.002 2.00275 11.835 11.835 5.259 5.259 2.167 2.167100 13.861 13.861 5.892 5.892 2.293 2.293250 22.989 22.989 8.478 8.478 2.750 2.750500 33.752 33.752 11.176 11.176 3.157 3.157750 42.265 42.265 13.140 13.140 3.423 3.4231000 49.581 49.581 14.741 14.741 3.625 3.625

Table 7. The comparison of the analytic results for −F ′′(0) given by the proposed homotopyanalysis method (HAM) when κ = 4/5, κ = 3/2 and κ = 4 with those given by the analyticformula (81).

converges quickly to the exact analytic result F ′′(0) = −√

2. The corresponding 10th-order approximation of F ′(ξ ) agrees well with the exact analytic solution F ′(ξ ) =exp(−

√2 ξ ). Then, we reconsider the case of κ = 2 and M = 1 and it is found that

the analytic approach proposed in this section gives a convergent result (with λ = 5/4and � = −4/7), and the homotopy-Pade approximant of F ′′(0) converges quickly tothe numerical result −1.187 given by Andersson et al. (1992). Finally, we considerthe case of M = 1 and κ = 4/5, a real power-law index. The convergent results areobtained when λ = 3/2 and � = −8/17, and the homotopy-Pade approximant ofF ′′(0) converges quickly to the numerical result −1.544 given by the Andersson et al.(1992), as shown in table 6. The corresponding 10th- and 20th-order approximationsof F ′(ξ ) agree well with the numerical result given by the Runge–Kutta method withF ′′(0) = −1.5441, as shown in figure 7. All of these verify the validity of the analyticapproach for any real power-law index κ proposed in this section.

In most cases, convergent results can be obtained by choosing λ and � as definedby (82) and (83), although, for M less or a little more than 2, a negative value of �had to be chosen closer to zero. All of our computations verify that

(a) the convergence regions can be enlarged as the auxiliary parameter � tends tozero from below;

(b) the convergence rate and region of F ′′(0) are greatly increased by the homotopy-Pade technique.The analytic results for F ′′(0) given by the proposed approach in a large range0 � M � 1000 for the integer power-law index κ = 4 and the real power-law indexκ = 4/5 and κ = 3/2 are listed in table 7, compared with those given by the formula(81). It is interesting that the analytic formula (81) gives good approximations ofF ′′(0) even for real power-law index κ , as shown in figure 8. So, we are quite sure that(81) is an accurate approximation of the dimensionless velocity gradient F ′′(0) forreal power-law index κ > 0 and magnetic field parameter 0 < M < +∞. Thus, using

210 S.-J. Liao

100

M

–F ��(0)

101 102 103

10

� = 4/5

� = 3/2

� = 4

20

30

40

50

Figure 8. Comparison of F ′′(0) given by (81) with the numerical results. Solid line: resultgiven by (81) when κ = 4/5; dashed line: result given by (81) when κ = 3/2; dash-dotted line:result given by (81) when κ = 4; open circle: the numerical solution when κ = 4/5; square:the numerical solution when κ = 3/2; triangle: the numerical solution when κ = 4.

(81) and the definition (10), we can calculate the skin friction on the sheet accuratelyenough for practical applications in engineering.

Based on numerical results for 0.4 � κ � 2 and 0 � M � 2, Andersson et al.(1992) found that −F ′′(0) increases monotonically with M for a given fluid butdecreases monotonically with κ for a given magnetic field. The analytic expression(81) indicates that this finding can be expanded to include non-Newtonian fluids withany real power-law index κ > 0 and any magnetic field parameter 0 < M < +∞, asshown by figures 6 and 8. Physically, it indicates that the effect of the non-Newtoniannature of the fluid is to reduce the skin friction, and that the effect of the magneticfield is to increase the skin friction.

4. Discussion and conclusionIn this paper the homotopy analysis method is employed to give analytic solutions

of the magnetohydrodynamic viscous flows of non-Newtonian fluids over a stretchingsheet. For the so-called second-order (κ = 2) and third-order (κ = 3) power-lawfluids the explicit analytic solutions are expressed by recursive formulas of constantcoefficients, which can be regarded as the definition of the solution of the nonlineardifferential equations (8) and (9) when κ = 2 and κ = 3. Also, for real power-lawindex κ > 0 and magnetic field parameter M in a quite large range we propose an


analytic approach. All of our analytic results agree well with the numerical ones. Inparticular, a rather simple analytic formula for the dimensionless velocity gradient atthe wall is found, which is quite accurate for real power-law index κ > 0 and magneticfield parameter 0 < M < +∞. This analytic formula (81) can give an accurate enoughskin friction result on the moving sheet to find wide applications in the metallurgyindustry. Physically, it indicates that the magnetic field tends to increase the wallfriction, and that this effect is more pronounced for shear-thinning (κ < 1) than forshear-thickening (κ > 1) fluids.

This paper also verifies that the homotopy analysis method is valid for complicated,strongly nonlinear problems in fluid mechanics.

Thanks to the editor and anonymous reviewers for their valuable suggestions. Thiswork is supported by National Science Fund for Distinguished Young Scholars ofChina (Approval No. 50125923).

REFERENCES

Adomian, G. 1976 Nonlinear stochastic differential equations. J. Math. Anal. Applic. 55, 441–452.

Adomian, G. 1994 Solving Frontier Problems of Physics: The Decomposition Method. Kluwer.

Alexander, J. C. & Yorke, J. A. 1978 The homotopy continuation method: numericallyimplementable topological procedures. Trans. Am. Math. Soc. 242, 271–284.

Andersson, H. I. 1992 MHD flow of a viscous fluid past a stretching surface. Acta Mechanica 95,227–230.

Andersson, H. I. 1995 A exact solution of the Navier-Stokes equations for magnetohydrodynamicsflow. Acta Mechanica 113, 241–244.

Andersson, H. I., Bech, K. H. & Dandapat, B. S. 1992 Magnetohydrodynamic flow of a power-lawfluid over a stretching sheet. Intl J. Non-Linear Mech. 27, 929–936.

Andersson, H. I. & Dandapat, B. S. 1991 Flow of a power-law fluid over a stretching sheet.Stability Appl. Anal. Continuous Media 1, 339–347.

Awrejcewicz, J., Andrianov, I. V., & Manevitch, L. I. 1998 Asymptotic Approaches in NonlinearDynamics . Springer.

Chakrabarti, A. & Gupta, A. S. 1979 Hydromagnetic flow and heat transfer over a stretchingsheet. Q. Appl. Maths 37, 73–78.

Chan, T. F. C. & Keller, H. B. 1982 Arc-length continuation and multi-grid techniques fornonlinear elliptic eigenvalue problems. SIAM J. Sci. Statist. Comput. 3, 173–193.

Cole, J. D. 1958 Perturbation Methods in Applied Mathematics . Blaisdell.

Crane, L. J. 1970 Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645–647.

Dandapat, B. S. & Gupta, A. S. 1989 Flow and heat transfer in a viscoelastic fluid over a stretchingsheet. Intl J. Non-Linear Mech. 24, 215–219.

Dinar, N. & Keller, H. B. 1985 Computations of taylor vortex flows using multigrid continuationmethods. Tech. Rep. California Institute of Technology.

Djukic, D. S. 1973 On the use of Crocco’s equation for the flow of power-law fluids in a transversemagnetic field. AIChE J. 19, 1159–1163.

Djukic, D. S. 1974 Hiemenz magnetic flow of power-law fluids. Trans. ASME: J. Appl. Mech. 41,822–823.

Erickson, L. E., Fan, L. T. & Fox, V. G. 1966 Heat and mass transfer on a moving continuous flatplate with suction or injection. Ind. Engng Chem. Fundam. 5, 19–25.

Grigolyuk, E. E. & Shalashilin, V. I. 1991 Problems of Nonlinear Deformation: The ContinuationMethod Applied to Nonlinear Problems in Solid Mechanics . Kluwer.

Hilton, P. J. 1953 An introduction to homotopy theory . Cambridge University Press.

Hinch, E. J. 1991 Perturbation Methods . Cambridge University Press.

Karmishin, A. V., Zhukov, A. I. & Kolosov, V. G. 1990 Methods of dynamics calculation and testingfor thin-walled structures . Mashinostroyenie, Moscow (in Russian).

212 S.-J. Liao

Liao, S. J. 1992 The proposed homotopy analysis technique for the solution of nonlinear problems.PhD thesis, Shanghai Jiao Tong University.

Liao, S. J. 1999a An explicit, totally analytic approximation of Blasius viscous flow problems. IntlJ. Non-Linear Mech. 34, 759–778.

Liao, S. J. 1999b A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flatplate. J. Fluid Mech. 385, 101–128.

Liao, S. J. 2002 An analytic approximation of the drag coefficient for the viscous flow past a sphere.Intl J. Non-Linear Mech. 37, 1–18.

Liao, S. J. 2003a An analytic approximate technique for free oscillations of positively dampedsystems with algebraically decaying amplitude. Intl J. Non-Linear Mech. 38, 1173–1183.

Liao, S. J. 2003b An explicit analytic solution to the Thomas-Fermi equation. Appl. Maths Comput.(to appear).

Liao, S. J. & Campo, A. 2002 Analytic solutions of the temperature distribution in Blasius viscousflow problems. J. Fluid Mech. 453, 411–425.

Liao, S. J. & Cheung, K. F. 2003 Homotopy analysis of nonlinear progressive waves in deep water.J. Engng Maths 45, 105–116.

Liao, S. J. & Chwang, A. T. 1998 Application of homotopy analysis method in nonlinear oscillations.Trans. ASME: J. Appl. Mech. 65, 914–922.

Lyapunov, A. M. 1892 General problem on stability of motion . English transl. Taylor & Francis,London, 1992.

McCormack, P. D. & Crane, L. J. 1973 Physical Fluid Dynamics . Academic.

McLeod, J. B. & Rajagopal, K. R. 1987 On the uniqueness of flow of a Navier-Stokes fluid dueto a stretching boundary. Arch. Rat. Mech. Anal. 98, 385–393.

Murdock, J. A. 1991 Perturbations: Theory and Methods . John Wiley & Sons.

Nayfeh, A. H. 2000 Perturbation Methods . John Wiley & Sons.

Pavlov, K. B. 1974 Magnetohydrodynamic flow of an impressible viscous fluid caused bydeformation of a surface. Magnitnaya Gidrodinamika 4, 146–147.

Rajagopal, K. R., Na, T. Y. & Gupta, A. S. 1984 Flow of a viscoelastic fluid over a stretchingsheet. Rheol. Acta 24, 213–215.

Rajagopal, K. R., Na, T. Y. & Gupta, A. S. 1987 A non-similar boundary layer on a stretchingsheet in a non-Newtonian fluid with uniform free stream. J. Math. Phys. Sci. 21, 189–200.

Sakiadis, B. C. 1961 Boundary layer behavior on continuous solid surface. AIChE. J. 7, 26–28.

Sarpkaya, T. 1961 Flow of non-Newtonian fluids in a magnetic field. AIChE. J. 7, 324–328.

Takhar, H. S., Raptis, A. A. & Perdikis, A. A. 1987 MHD asymmetric flow past a semi-infinitemoving plate. Acta Mechanica 65, 287–290.

Troy, W. C., Overman, E. A., Ermentrout, G. B. & Keener, J. P. 1987 Uniqueness of flow of asecond-order fluid past a stretching surface. Q. Appl. Math. 44, 753–755.

Vajravelu, K. 1986 Hydromagnetic flow and heat transfer over a continuous, moving, porous, flatsurface. Acta Mechanica 64, 179–185.

Van Dyke 1975 Perturbation Methods in Fluid Mechanics . The Parabolic Press.

Wang, C., Zhu, J. M., Liao, S. J. & Pop, I. 2003 On the explicit analytic solution of Cheng-Changequation. Intl J. Heat Mass Transfer 46, 1855–1860.

on the analytic solution of magnetohydrodynamic flows of non-newtonian fluids over a stretching...

Documents